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AP®MathsCollege BoardCalculus BCExam QuestionsUnit 1: Limits & ContinuityContinuity

Continuity (College Board AP® Calculus BC): Exam Questions

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1
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Let f be a function defined by f open parentheses x close parentheses equals open curly brackets table row cell sin open parentheses pi x close parentheses end cell cell for space x less than 1 end cell row cell ln x end cell cell for space x greater or equal than 1 end cell end table close.

Show that f is continuous at x equals 1.

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2
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Let f be the function defined by

f open parentheses x close parentheses equals open curly brackets table row cell 2 x minus 1 end cell cell 1 less or equal than x less or equal than 4 end cell row cell x squared minus 10 end cell cell 4 less than x less or equal than 7 end cell end table close

Is f continuous at x equals 4? Explain why or why not.

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3
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Let f be a continuous function such that f open parentheses 1 close parentheses equals 6 and f open parentheses 3 close parentheses equals 2.

Let g open parentheses x close parentheses equals f open parentheses x close parentheses plus x squared. Explain why there must be a value r for 1 less than r less than 3 such that g open parentheses r close parentheses equals 10.

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4
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bold italic x

0

4

7

9

12

Error converting from MathML to accessible text.

-6

-3

1

2

-5

f is a twice-differentiable function. The table above gives selected values of f to the power of apostrophe open parentheses x close parentheses over the time interval 0 less or equal than x less or equal than 12.

Is there a value of x, 4 less or equal than x less or equal than 7, for which f to the power of apostrophe open parentheses x close parentheses equals negative 1? Justify your answer.

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5
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Let f be the function defined by

f open parentheses x close parentheses equals open curly brackets table row cell fraction numerator x squared minus 3 x plus 2 over denominator x minus 2 end fraction end cell cell for space x not equal to 2 end cell row k cell for space x equals 2 end cell end table close

Given that f is continuous at every point, find the value of k.

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1
Sme Calculator2 marks

Let f be the function defined by

f open parentheses x close parentheses equals open curly brackets table row cell fraction numerator pi open parentheses x plus 2 close parentheses over denominator 2 end fraction space space space space for space x less than negative 1 end cell row cell arcsin open parentheses x squared close parentheses space space space for space minus 1 less or equal than x less or equal than 1 end cell end table close

Is f continuous at x equals negative 1? Use the definition of continuity to explain your answer.

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2a
Sme Calculator1 mark

Let f be the function defined by f open parentheses x close parentheses equals fraction numerator x squared minus 3 x over denominator x squared minus 2 x minus 3 end fraction.

Show that f has a removable discontinuity at x equals 3.

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2b
Sme Calculator1 mark

Explain how the discontinuity identified in part (a) can be removed.

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3
Sme Calculator1 mark

Let f be the function defined by f open parentheses x close parentheses equals open curly brackets table row cell 3 plus fraction numerator 7 over denominator x minus 4 end fraction space space space for space x less than 4 end cell row cell x minus 2 space space space space space space space space space space space for space x greater or equal than 4 end cell end table close.

Show that f has an essential discontinuity at x equals 4.

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4
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The value of a car manufactured by Warm Wheels is modeled by the continuous function W and the value of a car manufactured by Tommy Thunder is modeled by the function T. W open parentheses t close parentheses and T open parentheses t close parentheses are measured in US dollars and t is measured in years for 0 less or equal than t less or equal than 6.

Initially, the value of the car manufactured by Warm Wheels is $40,000 and after 6 years it is worth $15,500.The function T is given by T open parentheses t close parentheses equals 35000 times open parentheses 10 over 11 close parentheses to the power of t.

Let D open parentheses t close parentheses equals W open parentheses t close parentheses minus T open parentheses t close parentheses. Apply the Intermediate Value Theorem to the function D on the interval 0 less or equal than t less or equal than 6 to justify that there exists a time t, 0 less than t less than 6, at which both cars have the same value.

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5
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t (hours)

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8

12

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24

s open parentheses t close parentheses (ft)

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43

62

54

37

An animal is released into the wild. During the time interval 0 less or equal than t less or equal than 24 hours, the animal's distance from the place where it was released, measured in feet, is given by the differentiable function s. The table above shows selected values of this function.

For 0 less than t less than 24, must there be a time t when the animal is exactly 55 feet from the place where it was released? Justify your answer.

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1
Sme Calculator1 mark

Functions f and g are differentiable functions with f open parentheses 1 close parentheses equals g open parentheses 1 close parentheses equals 7. It is known that f open parentheses x close parentheses less or equal than g open parentheses x close parentheses for 0 less than x less than 2.

Let h be a function satisfying f open parentheses x close parentheses less or equal than h open parentheses x close parentheses less or equal than g open parentheses x close parentheses for 0 less than x less than 2. Is h continuous at x equals 1? Justify your answer.

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2
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t (seconds)

0

1

5

10

B open parentheses t close parentheses (square inches / second)

5

9

31

82

A balloon is inflated with air using a machine. The surface area of the balloon is increasing at a rate modeled by A open parentheses t close parentheses equals 4 e to the power of bevelled t over 3 end exponent square inches per second for 0 less or equal than t less or equal than 10. A second balloon is inflated with air by a human. The surface area of the second balloon is increasing at a rate modeled by B open parentheses t close parentheses square inches per second, where B is differentiable on 0 less or equal than t less or equal than 10. Selected values of B open parentheses t close parentheses are shown in the table above.

For 0 less or equal than t less or equal than 10, is there a time t when the rate at which the surface areas of both balloons are increasing at the same rate? Explain why or why not.

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3
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Let f be the function defined by

f open parentheses x close parentheses equals open curly brackets table row cell 2 x plus 7 end cell cell for space x less or equal than a end cell row cell fraction numerator x cubed minus x squared minus 3 x minus 9 over denominator x squared minus 4 x plus 3 end fraction end cell cell for space a less than x less than 3 end cell row cell x to the power of b end cell cell for space x greater or equal than 3 end cell end table close.

Given that f is continuous for all real values of x, find the value a and the value of b.

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4
Sme Calculator2 marks

x

space f open parentheses x close parentheses space

space g open parentheses x close parentheses space

1

-5

1

3

1

5

5

-10

7

7

7

4

The functions f and g are continuous functions. The table above gives values of the functions at selected values of x. The function h is given by h open parentheses x close parentheses equals open vertical bar f open parentheses g open parentheses x close parentheses close parentheses close vertical bar.

For 1 less than x less than 5, what is the minimum number of solutions that there must be to the equation h open parentheses x close parentheses equals 8? Justify your answer.

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5a
Sme Calculator2 marks

Let f be the function defined by

f open parentheses x close parentheses equals open curly brackets table row cell ln open parentheses x squared minus 3 close parentheses end cell cell for space x less or equal than negative 2 end cell row cell fraction numerator 2 x squared minus 10 x plus 8 over denominator x squared minus 2 x minus 8 end fraction end cell cell for space minus 2 less than x less than 4 end cell row cell tan open parentheses pi over 2 minus pi over x close parentheses end cell cell for space x greater than 4 end cell end table close.

Explain whether f is continuous at x equals negative 2. If f has a discontinuity at this point, then describe the type of discontinuity.

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5b
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Explain whether f is continuous at x equals 4. If f has a discontinuity at this point, then describe the type of discontinuity.

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